## Modules 2017–18

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UG code: MTH5104
Module name:
Description:

This module introduces some of the mathematical theory behind Calculus. It answers questions such as: What properties of the real numbers do we rely on in Calculus? What does it mean to say that a series converges to a limit? Are there kinds of function that are guaranteed to have a maximum value? The module is a first introduction, with many examples, to the beautiful and important branch of pure mathematics known as Analysis.

Organiser: Dr Xin Li*
Details: Level |Semester A|15 credits
Subject area: Pure mathematics
Overlaps:

None

Essential prerequisites:
• MTH4100/MTH4200 Calculus I
• MTH4104 Introduction to Algebra

None

Restrictions:

None

Assessment:

#### Normal assessment

DescriptionDuration / LengthWeighting
Mid-term test 40 minutes 10%
Final examination 2 hours 90%

#### Associate assessment

DescriptionDuration / LengthWeighting
Mid-term test 40 minutes 10%
Final examination 2 hours 90%

#### Reassessment – synoptic

DescriptionDuration / LengthWeighting
Resit examination 2 hours 100%
Syllabus:
1. Real numbers: Algebraic properties, inequalities, supremum and infimum, completeness axiom for the existence of the supremum.
2. Sequences: Definition of limit and its use in specific examples, limit of sum, product and quotient of sequences. Bounded monotone sequences. Statement of the Bolzano-Weierstrass Theorem.
3. Series: Convergent series, geometric series, harmonic series. Comparison and ratio tests. Absolutely convergent series. Power series. Examples, including sin(x), cos(x) and exp(x).
4. Continuous functions: Definition of continuity and its use in specific examples, sum of continuous functions, composites of continuous functions (proofs), products/quotients of continuous functions (stated). Briefly, the Intermediate Value Theorem, application to roots of polynomials, boundedness of continuous functions on closed bounded intervals.
5. Definition of derivative. Continuity of differentiable functions. (Covered if time permits, also covered at the start of the follow-on module, Differential and Integral Analysis.)
Learning outcomes:

This module covers:

• The fundamental concepts of mathematical analysis.
• The formal definitions of convergence for sequences and series, and of continuity and differentiability for functions.
• Proofs of general results which follow from these definitions.

#### Disciplinary Skills

At the end of this module, students should be able to:

• Define the basic concepts underlying continuous mathematics: supremum, limit of a sequence, convergent series, continuous function, derivative.
• Use criteria for convergence of series and continuity of functions.
• Prove general results concerning infinite sequences, convergent series and continuous functions.
• Solve problems relating to convergence of series and continuity of functions.

#### Attributes

At the end of this module, students should have developed with respect to the following attributes:

• Grasp the principles and practices of their field of study.
• Explain and argue clearly and concisely.
• Apply their analytical skills to investigate unfamiliar problems.
• Acquire substantial bodies of new knowledge.

#### Assessment Criteria

History:

Approved by Faculty Board 03/99. CD and TA added, level revised 05/03, assessment profile changed 06/04. Assessment in 2010–11 was 10% coursework, 10% in-term test, 80% final exam.